Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the classifica
tion proves their existence, it does not indicate a way of calculating the values of those invariants. Here, we show that a specific set of concentric Wilson loops and line invariants yields the values of all topological invariants in two-dimensional systems with pure rotation symmetry in class AII. Examples of this analysis are given for specific models with two-fold and three-fold rotational symmetry. We find new invariants that correspond to the presence of higher-order topology and corner charges.
Thinning crystalline materials to two dimensions (2D) creates a rich playground for electronic phases, including charge, spin, superconducting and topological order. Bulk materials hosting charge density waves (CDWs), when reduced to ultrathin films,
have shown CDW enhancement and tunability. However, charge order confined to only 2D remains elusive. Here we report a distinct charge ordered state emerging in the monolayer limit of 1T-VSe$_2$. Systematic scanning tunneling microscopy experiments reveal that bilayer VSe$_2$ largely retains the bulk electronic structure, hosting a tri-directional CDW. However, monolayer VSe$_2$ exhibits a dimensional crossover, hosting two CDWs with distinct wavelengths. Electronic structure calculations reveal that while one CDW is bulk-like and arises from the well-known Peierls mechanism, the other is decidedly unconventional. The observed CDW-lattice decoupling and the emergence of a flat band suggest that the new CDW arises from enhanced electron-electron interactions in the 2D limit. These findings establish monolayer-VSe$_2$ as the first host of coexisting charge orders with distinct origins, opening the door to tailoring electronic phenomena via emergent interactions in 2D materials.
We present a detailed study of the bulk electronic structure of high quality VSe$_{2}$ single crystals using optical spectroscopy. Upon entering the charge density wave phase below the critical temperature of 112 K, the optical conductivity of VSe$_2
$ undergoes a significant rearrangement. A Drude response present above the critical temperature is suppressed while a new interband transition appears around 0.07,eV. From our analysis, we estimate that part of the spectral weight of the Drude response is transferred to a collective mode of the CDW phase. The remaining normal state charge dynamics appears to become strongly damped by interactions with the lattice as evidenced by a mass enhancement factor m$^{*}$/m$approx$3. In addition to the changes taking place in the electronic structure, we observe the emergence of infrared active phonons below the critical temperature associated with the 4a x 4a lattice reconstruction.
